General Prerequisites: Part A Differential Equations 1; A7 Numerical Analysis is desirable but not essential.
Course Overview: To introduce and give an understanding of numerical methods for the solution of elliptic, parabolic and hyperbolic partial differential equations, including their derivation, analysis and applicability.
Learning Outcomes: At the end of the course the student will be able to:
Construct practical methods for the numerical solution of boundary-value problems arising from ordinary differential equations and elliptic partial differential equations; analyse the stability, accuracy, and uniqueness properties of these methods; construct methods for the numerical solution of initial-boundary-value problems for second-order parabolic partial differential equations, and first- and second-order hyperbolic partial differential equations, and analyse their stability and accuracy properties.
Course Synopsis: The course is devoted to the development and analysis of numerical approximations to boundary-value problems for second-order ordinary differential equations, boundary-value problems for second-order elliptic partial differential equations, initial-boundary-value problems for second-order parabolic equations, and first- and second-order hyperbolic partial differential equations. The course begins by considering classical techniques for the numerical solution of boundary-value problems for second-order ordinary differential equations and elliptic boundary-value problems, in particular the Poisson equation in two dimensions. Topics include: discretisation, stability and convergence analysis, and the use of the discrete maximum principle. The remaining lectures focus on the numerical solution of initial-boundary-value problems for second-order parabolic and first- and second-order hyperbolic partial differential equations with topics such as: approximation by finite difference methods, accuracy, stability (including the Courant-Friedrichs-Lewy (CFL condition) and convergence.
